25.5 Fractional and Decimal Linear Inequalities

Fractional and Decimal Linear Inequalities are one-variable linear inequalities containing fractional or decimal coefficients and constants, requiring the same clearing techniques established for fractional and decimal equations, but with the added requirement of tracking the sign of any multiplier used, since a negative multiplier applied to clear such an inequality triggers Relation Reversal after Negative Scaling.

Fractional Inequality Coefficient Handling addresses the case in which the variable in an inequality carries a fractional coefficient, requiring either Reciprocal Operation for Variable Isolation, using the reciprocal of that fraction, or Numerical Denominator Clearing applied to the entire inequality, with either approach subject to the same sign-tracking discipline required of every multiplicative inequality transformation.

Positive Common Multiplier Selection is the recommended refinement of Common Equation Multiplier Selection when applied to an inequality: whenever possible, the common multiplier chosen to clear every denominator in the inequality should be selected as a positive value, since every denominator identified through Equation Denominator Inventory is, by convention, expressed as a positive number, and choosing a positive common multiplier avoids the need for Relation Reversal after Negative Scaling during the clearing step itself.

Decimal Scaling by a Positive Power of Ten is the corresponding refinement of Decimal Clearing by Scaling when applied to an inequality: because a Power-of-Ten Equation Multiplier is by its nature always a positive value, scaling a decimal inequality by an appropriate power of ten never requires a symbol reversal at the scaling step, making this particular clearing technique inherently safe with respect to Comparison Direction Preservation.

Sign Check before Numerical Scaling is the mandatory verification that must precede any multiplication or division applied to a fractional or decimal inequality, whether during a preliminary clearing step following Positive Common Multiplier Selection and Decimal Scaling by a Positive Power of Ten, or during Final Inequality Coefficient Step once the inequality has been reduced to a single coefficient. This check determines definitively whether the quantity about to be used is positive or negative, and therefore whether Positive Scaling without Relation Reversal or Relation Reversal after Negative Scaling governs the resulting inequality symbol.

Exact Solution Boundary Preservation is the requirement that the boundary value produced by solving a fractional or decimal inequality be carried forward in its full, exact form, whether as a reduced fraction or as an exact terminating or repeating decimal, following the same standard established by Exact Fractional Solution Form and Decimal Result and Exactness Check for equations. Because the boundary value determines the precise location of Inequality Boundary Placement in any graphical or interval-based representation of the solution, an approximated or rounded boundary would misrepresent the true extent of the solution set, even if the direction of the inequality itself were correctly determined.